Question:
Two rods of equal length \(60\,\text{cm}\) each are joined together end to end. The coefficients of linear expansion of the rods are \(24\times10^{-6}\^{\circ}\text{C}^{-1}\) and \(1.2\times10^{-5}\^{\circ}\text{C}^{-1}\). Their initial temperature is \(30^{\circ}\text{C}\), which is increased to \(100^{\circ}\text{C}\). Find the final length of the combination (in cm).
Two rods of equal length \(60\,\text{cm}\) each are joined together end to end. The coefficients of linear expansion of the rods are \(24\times10^{-6}\^{\circ}\text{C}^{-1}\) and \(1.2\times10^{-5}\^{\circ}\text{C}^{-1}\). Their initial temperature is \(30^{\circ}\text{C}\), which is increased to \(100^{\circ}\text{C}\). Find the final length of the combination (in cm).
Show Hint
For thermal expansion of composite rods:
Calculate expansion of each rod separately
Use \( \Delta L = L\alpha\Delta T \)
Add individual expansions to get total increase
Calculate expansion of each rod separately
Use \( \Delta L = L\alpha\Delta T \)
Add individual expansions to get total increase
Updated On: Jan 31, 2026
- \(120.1321\)
- \(120.1123\)
- \(120.1512\)
- \(120.1084\)
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The Correct Option is C
Solution and Explanation
Step 1: Given:
\[
L_1 = L_2 = 60\,\text{cm},
\alpha_1 = 24\times10^{-6},
\alpha_2 = 12\times10^{-6}
\]
\[
\Delta T = 100 - 30 = 70^{\circ}\text{C}
\]
Step 2: Linear expansion of first rod: \[ \Delta L_1 = L_1 \alpha_1 \Delta T = 60 \times 24\times10^{-6} \times 70 = 0.1008\,\text{cm} \]
Step 3: Linear expansion of second rod: \[ \Delta L_2 = L_2 \alpha_2 \Delta T = 60 \times 12\times10^{-6} \times 70 = 0.0504\,\text{cm} \]
Step 4: Total increase in length: \[ \Delta L = 0.1008 + 0.0504 = 0.1512\,\text{cm} \]
Step 5: Final length of the combination: \[ L_{\text{final}} = 120 + 0.1512 = 120.1512\,\text{cm} \]
Step 2: Linear expansion of first rod: \[ \Delta L_1 = L_1 \alpha_1 \Delta T = 60 \times 24\times10^{-6} \times 70 = 0.1008\,\text{cm} \]
Step 3: Linear expansion of second rod: \[ \Delta L_2 = L_2 \alpha_2 \Delta T = 60 \times 12\times10^{-6} \times 70 = 0.0504\,\text{cm} \]
Step 4: Total increase in length: \[ \Delta L = 0.1008 + 0.0504 = 0.1512\,\text{cm} \]
Step 5: Final length of the combination: \[ L_{\text{final}} = 120 + 0.1512 = 120.1512\,\text{cm} \]
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